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Calculus. Limits Continuity of Functions Differentiation Application of differentiation Integration. Limits. Example from tangent line Example from Decimal expansion of 1/3=0.33333…. Definition of limits – get closer but do not touch Few more examples of limits. Limit of series.
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Calculus • Limits • Continuity of Functions • Differentiation • Application of differentiation • Integration
Limits • Example from tangent line • Example from Decimal expansion of 1/3=0.33333…. • Definition of limits – get closer but do not touch • Few more examples of limits
Limits • Example from geometric series • Limit of a series • Definition of limits – get closer but do not touch • Few more examples of limits
Some Important Limits • sin(x)/x at x=0 • Definition of e • Limit of 1/x at x=0 • Limit of 1/(x*x) at x=0 • more examples
A practical Example • Interest compounded continuously • S = P[(1 + r/k)^(kn)] where n– no of years, k is no of times interest is compounded in a year, other variables have usual meaning. • Let k -> infinity. • S = P* e^(rn) • E.g. a trust fund is set by a single payment so that at the end of 20 years, there is $25000 in the fund. If the interest is calculated at a rate of 7% compounded continuously, find the amount of money to be invested.
Continuity • Definition • f(x)=1/x if x‚0; is f(x) continuous at x=0? • f(x)=1/x if x‚0 otherwise f(x)=0; is f continuous at x=0? • f(x)=x^2/x if x‚0 otherwise f(x)=1; is f continuous at x=0? • f(x)=x^2/x if x‚0 otherwise f(x)=0; is f continuous at x=0?
Differentiation • Algebraic motivation – rate of change
Differentiation • Algebraic motivation – rate of change • Geometric motivation – limiting position of secant
Differentiation • Definition • Rules of Differentiation • Examples • Differentiable Functions
Some more Differentiation • Derivatives of Logarithmic functions • Derivatives of Exponential functions • Derivative of implicit functions • examples • Higher Order Derivatives
Multivariate Calculus • Functions of Several Variables • E.g. The marketing quantity may depend on the own price and price of competition. • Represented in a 3-D plane rather than a line. • Important in complex dependency models
Partial Differentiation • Rate of change of function when one variable changes • example
Applications of Partial Derivatives • E.g. A company manufactures two types of products A and B. Suppose the joint cost function for producing x pairs of A and y pairs of B is • C = 0.06x*x + 65x + 75y + 1000 • Determine the marginal costs wrt x and y and evaluate these when x = 100 and y = 50. Interpret.
Applications of Partial Derivatives • E.g. MBA compensation : In a study of success among MBAs, it was calculated that current annual compensation was given by • Z = 10,990 + 1120x + 873y, where x and y are number of years of work ex prior and post MBA. Find the partial derivative wrt x and y and interpret.
Higher Order Partial Derivatives • When a function depends on two variables and we desire to find the sensitivity wrt both at the same time • Derivative can be taken wrt same variable more than once also to determine the nature of the rate of change • example
Application of Differentiation • Extremal Problems – maxima and minima • Curve Sketching • Mathematical Modelling